Av(1324, 1342, 1423, 2143)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{4} \sqrt{-4 x +1}+3 x^{4}-8 x^{3}+12 x^{2}-6 x +1}{2 x^{3} \left(x^{2}-2 x +2\right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 68, 234, 815, 2870, 10204, 36582, 132103, 480094, 1754654, 6445278, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{3} \left(x^{2}-2 x +2\right) F \left(x \right)^{2}+\left(-3 x^{4}+8 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{\left(1+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(6 n +31\right) a \! \left(3+n \right)}{7+n}+\frac{\left(13 n +28\right) a \! \left(n +1\right)}{14+2 n}-\frac{\left(59+19 n \right) a \! \left(n +2\right)}{2 \left(7+n \right)}, \quad n \geq 4\)
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This specification was found using the strategy pack "Point Placements" and has 22 rules.

Found on July 23, 2021.

Finding the specification took 3 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\ \end{align*}\)